Question

the graph of f(x), shown below, resembles the graph of g(x) = x², but it has been changed somewhat. which of the following could be the equation of f(x)? a. f(x) = 3(x - 2)² - 2 b. f(x) = -3(x + 2)² - 2 c. f(x) = -3(x - 2)² - 2

Explanation:

1. Recall the vertex form of a parabola: \( F(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the direction (up/down) and vertical stretch/compression.
2. Analyze the graph of \(F(x)\) compared to \(G(x)=x^2\) (which opens upward, vertex at \((0,0)\)):

• \(F(x)\) opens downward (so \(a < 0\)), eliminating option A (where \(a = 3>0\)).
• The vertex of \(F(x)\): From the graph, it appears to be at \((-2, -2)\) (since it's shifted left and down from the origin).
• For option B: \(F(x)=-3(x + 2)^2 - 2\) has \(h=-2\) (since \(x - (-2)=x + 2\)), \(k=-2\), and \(a=-3<0\) (opens downward), matching the vertex and direction.
• For option C: \(F(x)=-3(x - 2)^2 - 2\) has vertex at \((2, -2)\), which does not match the graph’s vertex position (left of the y - axis), so C is incorrect.

Answer:

B. \( F(x) = -3(x + 2)^2 - 2 \)